Sequence And Series Question 415
Question: If $ a+b+c=3 $ and $ a>0,b>0,c>0, $ then the greatest value of $ a^{2}b^{3}c^{2} $ is
Options:
A) $ \frac{3^{10}{{.2}^{4}}}{7^{7}} $
B) $ \frac{3^{9}{{.2}^{4}}}{7^{7}} $
C) $ \frac{3^{8}{{.2}^{4}}}{7^{7}} $
D) None of these
Show Answer
Answer:
Correct Answer: A
Solution:
[a] Taking A.M. and G.M. of 7 numbers $ \frac{a}{2},\frac{a}{2},\frac{b}{3},\frac{b}{3},\frac{b}{3},\frac{c}{2},\frac{c}{2}, $ we get $ \frac{2.\frac{a}{2}+3.\frac{b}{3}+2.\frac{c}{2}}{7}\ge {{{ {{( \frac{a}{2} )}^{2}}{{( \frac{b}{3} )}^{3}}{{( \frac{c}{2} )}^{2}} }}^{\frac{1}{7}}} $
$ \Rightarrow ,\frac{3^{7}}{7^{7}}\ge \frac{a^{2}b^{3}c^{2}}{2^{2}{{.3}^{3}}{{.2}^{2}}},\Rightarrow a^{2}b^{3}c^{2}\le \frac{3^{10}{{.2}^{4}}}{7^{7}} $
$ \therefore $ greatest value of $ a^{2}b^{3}c^{2}=\frac{3^{10}{{.2}^{4}}}{7^{7}} $
BETA
